I want the text view to be scrollable as the text begins to fill the box, but the box keeps on expanding to accommodate more text. How to do I limit the size of the text view and create a scrollable text view.

I'm afraid you've misunderstood how scrollbars work in GTK; usually you don't create a scrollbar directly, but you place the widget you would like to scroll in a GtkScrolledWindow. This creates scrollbars automatically and connects them to the widget inside the scrolled window; in your case, the text view.